Calculating R-Square from Line Fit Equation- Tutorial

What R-Squared Actually Is

R-squared (R²) measures how well your regression line fits the data. It's the proportion of variance in the dependent variable explained by your independent variable(s). A value of 0.85 means 85% of the variation is explained by your model. The remaining 15% is noise you can't account for.

Most people calculate R² using software and never think about the math underneath. But when you have a line fit equation already, you can derive R² directly without rerunning your regression. Here's how.

The R-Squared Formula

The standard formula uses two sums of squares:

R² = 1 - (SSres / SStot)

Where:

• SSres = Sum of squared residuals (explained variation)
• SStot = Total sum of squares (total variation)

SStot is straightforward. It's the sum of squared differences between each y-value and the mean of all y-values:

SStot = Σ(yi - ȳ)²

SSres is the sum of squared differences between actual y-values and predicted y-values from your line:

SSres = Σ(yi - ŷi)²

Calculating R² from Your Line Fit Equation

Your line fit equation looks like this:

ŷ = mx + b

Where m is slope and b is intercept. Here's the step-by-step process:

Step 1: Gather Your Data

You need the original x and y data points. The line equation alone isn't enough—you need the actual y-values to calculate residuals.

Step 2: Calculate the Mean of Y

Add all your y-values and divide by n (number of data points). This is ȳ.

Step 3: Calculate SStot

For each data point, subtract the mean from the actual y-value, square it, and sum all results.

Step 4: Calculate SSres

For each x-value, plug it into your line equation to get the predicted ŷ. Subtract that from the actual y-value, square it, and sum all results.

Step 5: Plug Into the Formula

Divide SSres by SStot, subtract from 1, and you have R².

Worked Example

Say you have five data points and your line fit is ŷ = 2.5x + 10.

Mean of y (ȳ) = 20

SSres = 6.25 + 9 + 6.25 + 4 + 6.25 = 31.75

SStot = 25 + 64 + 100 + 144 + 225 = 558

R² = 1 - (31.75 / 558) = 1 - 0.0569 = 0.943

Your line explains 94.3% of the variance. That's a strong fit.

Quick Method: Correlation Coefficient

If your regression is simple linear (one x variable), you can skip the residuals calculation entirely. R² equals the square of the correlation coefficient (r):

R² = r²

Calculate r between your x and y values, square it, and you're done. This only works for simple linear regression with a single predictor.

What R² Can't Tell You

R² has limits. A high R² doesn't mean your model is correct—it means it fits the data you have. You could be overfitting. It also doesn't tell you if you chose the right functional form. A quadratic relationship forced into a linear model will give you a misleading R².

Low R² isn't always bad either. In noisy domains like economics or social science, R² of 0.3 might be meaningful. Context matters.

Adjusted R²

Adding more predictors always increases R², even if the new variables are useless. Adjusted R² penalizes you for adding predictors that don't contribute meaningful explanatory power. If you're comparing models with different numbers of variables, use adjusted R²:

Adjusted R² = 1 - [(1 - R²)(n - 1) / (n - k - 1)]

Where n is sample size and k is number of predictors.

Tools That Do This Automatically

• Excel/Google Sheets: =RSQ(y_range, x_range) gives R² directly
• Python: sklearn.metrics.r2_score(y_true, y_pred)
• R: summary(lm(y ~ x))$r.squared
• Online calculators**: Many free tools let you paste data and get R² in seconds